Short Edge Of Hexakis Icosahedron Given Total Surface Area Calculator

Formula Used:

\[ \text{Short Edge} = \frac{5}{44} \times (7 - \sqrt{5}) \times \sqrt{\frac{44 \times \text{Total Surface Area}}{15 \times \sqrt{10 \times (417 + 107 \times \sqrt{5})}}} \]

Short Edge of Hexakis Icosahedron:

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1. What is the Short Edge of Hexakis Icosahedron?

The Short Edge of Hexakis Icosahedron is the length of the shortest edge that connects two adjacent vertices of the Hexakis Icosahedron. It is a key geometric parameter in understanding the structure and properties of this complex polyhedron.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Short Edge} = \frac{5}{44} \times (7 - \sqrt{5}) \times \sqrt{\frac{44 \times \text{Total Surface Area}}{15 \times \sqrt{10 \times (417 + 107 \times \sqrt{5})}}} \]

Where:

\( \text{Total Surface Area} \) — Total surface area of the Hexakis Icosahedron (m²)
\( \sqrt{} \) — Square root function

Explanation: The formula calculates the shortest edge length based on the total surface area of the Hexakis Icosahedron, incorporating mathematical constants and geometric relationships specific to this polyhedron.

3. Importance of Short Edge Calculation

Details: Calculating the short edge is essential for understanding the geometric properties, symmetry, and spatial characteristics of the Hexakis Icosahedron. It is particularly important in fields such as crystallography, molecular modeling, and advanced geometry studies.

4. Using the Calculator

Tips: Enter the total surface area in square meters. The value must be positive and greater than zero. The calculator will compute the corresponding short edge length of the Hexakis Icosahedron.

5. Frequently Asked Questions (FAQ)

Q1: What is a Hexakis Icosahedron?
A: A Hexakis Icosahedron is a Catalan solid that is the dual of the truncated dodecahedron. It has 120 faces, 180 edges, and 62 vertices.

Q2: Why is this formula so complex?
A: The complexity arises from the intricate geometric relationships and mathematical constants involved in describing the properties of the Hexakis Icosahedron, which is a highly symmetric but complex polyhedron.

Q3: What are typical values for the short edge?
A: The short edge length depends on the size of the polyhedron. For a standard Hexakis Icosahedron with specific dimensions, typical values range from fractions of a meter to several meters, depending on the scale.

Q4: Can this calculator be used for other polyhedra?
A: No, this calculator is specifically designed for the Hexakis Icosahedron. Other polyhedra have different geometric relationships and require different formulas.

Q5: What precision should I expect from the results?
A: The calculator provides results with 6 decimal places precision, which is sufficient for most geometric and engineering applications involving the Hexakis Icosahedron.